Our MLFF for solid ZrO\(_2\)  was trained on the fly during MD simulations in the
NPT ensemble at ambient pressure, with the temperature and pressure controlled
by a Langevin thermostat [1]} combined with the Parrinello-Raman
method [2]}, [3]}. The training was performed on a
supercell containing 96 atoms using a time step of 2.5 fs. First, the system
was gradually heated from 1600 K to 2800 K for 180 ps, then it was heated again
starting from 500 K to 1600 K for another 180 ps. The initial structures for
these MD simulations were obtained by equilibrating the tetragonal and monoclinic
structures at 1600 K and 500 K, respectively, for 50 ps using the on-the-fly
scheme and discarding the MLFFs thus generated. Finally, to sample the
low-temperature, unstable cubic, and tetragonal structures as well, we performed 10
additional MD steps at 100 K starting from the DFT-relaxed structures. In total,
only 592 first-principles calculations were executed, out of almost 145,000 MD
steps. These constitute the reference structures dataset, from which \(N_b=831\) 
and 1090 local reference configurations were selected for Zr and O, respectively.
